This article is cited in 6 papers

Mathematical problems of nonlinearity

Rotation Number as a Complete Topological Invariant of a Simple Isotopic Class of Rough Transformations of a Circle

Nozdrinova E. V.

Laboratory of Topological Methods in Dynamics, NRU HSE, ul. Bolshaya Pecherskaya 25/12, Nizhny Novgorod, 603155 Russia

Abstract: The problem of the existence of a simple arc connecting two structurally stable systems on a closed manifold is included in the list of the fifty most important problems of dynamical systems. This problem was solved by S. Newhouse and M. Peixoto for Morse – Smale flows on an arbitrary closed manifold in 1980. As follows from the works of Sh. Matsumoto, P. Blanchard, V. Grines, E. Nozdrinova, and O. Pochinka, for the Morse – Smale cascades, obstructions to the existence of such an arc exist on closed manifolds of any dimension. In these works, necessary and sufficient conditions for belonging to the same simple isotopic class for gradient-like diffeomorphisms on a surface or a three-dimensional sphere were found. This article is the next step in this direction. Namely, the author has established that all orientation-reversing diffeomorphisms of a circle are in one component of a simple connection, whereas the simple isotopy class of an orientation-preserving transformation of a circle is completely determined by the Poincaré rotation number.

Keywords: rotation number, simple arc.

MSC: 37D15

Received: 05.11.2018
Accepted: 27.11.2018

Language: English

DOI: 10.20537/nd180408

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